harmonic metrics

I've been following the thread on harmonic spaces and metrics with
interest, and although it's been a while since I've done any work in
this area I'd like to comment briefly.

I've found it somewhat fruitful to think of any given harmonic metric --
like any given tuning -- as one way of hearing among many ways.

In fact I'm skeptical that there can be any such thing as a single
metric (in the strict mathematical sense) for "consonance" or
"concordance" or "harmonicity", because these concepts are so highly
contextual. I don't believe that ANY metric can state categorically
that a 15/1 is more or less "consonant" than a 5/3, or a 15/8 than a
16/15, or a root-position major triad than a second-inversion triad, or
any such comparison, without carefully circumscribing its assertion by a
precise definition of what "consonance" means in its context. (Tenney's
_History of 'Consonance' and 'Dissonance'_ is a valuable look at how
notions of "consonance" have changed over the centuries in European art
music.) In designing a metric, there is an understandable tendency to
shuttle between number and intuition, to say "my ears tell me..." about
this quality, while also striving for quantitative objectivity.

Which is NOT to say that the quality can't be measured, or that the
design of metrics is therefore futile. But it does seem inevitable that
any given metric will necessarily contain highly subjective decisions --
e.g. how to weight the various dimensions, whether or not it's octave
invariant, whether the limit-system is based on primes or odd numbers or
odd composites, et cetera -- decisions that carry with them theoretical
assumptions about the nature of harmony and tonality, tending to
radically simplify and somewhat falsify the processes by which we hear
and interpret.

I believe that a somewhat fuller and more useful description of
harmonicity might be reached by using a GROUP of metrics, each
descriptive of some particular aspect of pitch and harmonic space,
instead of seeking a single "best" metric for a quality as multivalent
and contextual as "consonance".

Various metrics based on small-number-ratio theories of consonance have
been discussed extensively on the list. I would like to see more
discussion of what I consider "problem areas", the first two of them
psychoacoustic, the third a more general agenda:

1) Critical bandwidth. Though there has been much discussion of octave
invariance, there has been little of critical bandwidth, which requires
that we bring absolute frequency, not just interval ratios, into any
metric that claims psychoacoustic validity. (Sethares's extension of
Plomp-Levelt does this, and takes steady-state timbre into account as
well.)

2) Non-integers/approximations. Small-number-ratio metrics fail with
irrationals. E.g. 300001/200000 is much more "dissonant" than 3/2;
indeed, the closer the approximation, the more distant it's ranked by
any small-number metric. The ear has a much more adaptable threshold.
Some metric that describes near-misses could be quite useful. (Again,
by incorporating critical bandwidth, Sethares's metric avoids this
particular pitfall. Tenney alludes to, but does not incorporate, a
"tolerance range" in his harmonic distance metric.)

3) Acculturation. A catch-all by which I mean any sort of bias or
assumption or preference that may be hiding behind a veneer of
objectivity. Octave-invariance has been discussed. Some discussion of
where & how qualities become quantities might be helpful, e.g. the
importance of 5-limit intervals in tertial harmony. Not that bias or
preference could or should be banished, but it ought to be identified.
Thus, a metric designed to describe the closeness of a pitch set to a
harmonic series might reasonably rank a 7/1 closer than a 3/2. Et cetera.

The goal might be to arrive at some set of simple metrics that do not
confuse a multi-dimensional subjective quality like "consonance" with a
single quantitative measurement. One could then choose the qualities
that one deems important and use those metrics appropriate to the
analytical or compositional task at hand.

I wrote:

>If the dissonance of a composite number is independent of its
>factors, why use lattices at all?

and Paul replied:

>What an odd question! What could you possibly mean by it?

Okay, the first part of my question is a condition. If you're
going to give a new direction to 9, you may also give one to 15,
21 and so on until every number has its own diection. Nobody has
suggested this, but then nobody has said why we should stop at
9 either.

Using the harmonic distance on a rectangular lattice as a measure
of dissonance is an economical theory. All octave invariant,
5-prime-limit intervals can be expressed in 2 dimensions on this
lattice. The dissonance of these intervals can be calculated
using 2 free parameters: the step lengths in the 3- and 5-
directions. The method can be generalised into any number of
prime dimensions with a new free parameter for every new
dimension. Switch to a triangular lattice, and you need n-1 new
free parameters for n-dimensional harmony, giving n(n-1) in total.
These parameters can all be produced from the same formula, like
log(max(m,n)).

Give each integer its own direction, though, and you need n free
parameters to assess n+1 intervals (including 1/1). You could
just as easily write down a list of the intervals you plan to use,
and say how dissonant each is. So, why use a lattice? Or, if 9
is special, why?


Actually, since I asked that original question something did
occur to me about these composite lattices. They are geting very
close to being a Hilbert space. I think this may be important,
but I will not explain myself here because I don't expect most
list subscribers will know what a Hilbert space is, let alone what its
significance might be.


Approximate scales where 9/1 approximates to be different to
twice 3/1 are an entirely different idea. Don't use arguments
for one to justify the other. There is nothing wrong with the
basic dimension of a scale changing with an approximation.
Incidentally, Paul, why do you think your previous comment on
this contradicts what I said?

All equal temperaments have a basic dimension of 1.

If 9 approximates to an odd number, how would you tune 1:3:9? If
3/1 and 9/1 are optimised, 9/3=3/1 must be wrong. Or do you plan
to avoid all chords where 9 and 3 both occur? Otherwise, you
*could* equally say that you have two different 3- directions.

This problem doesn't arise with harmonic dimensionality. You
could curve your triangular, 3-5 lattice and add extra links
between the 9's. This will be quantitatively the same as giving 9
a new dimension.



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